On the Maxwell Constants in 3D Dirk Pauly Fakultät für Mathematik Universität Duisburg-Essen, Campus Essen, Germany JSA 3 & MC 65 Journées Singulières Augmentés 03: Conférence en l honneur de Martin Costabel pour ses 65 ans August 7 03 Rennes, France
Sobolev Spaces Ω R 3 bounded domain with Lipschitz (or weaker) boundary Γ = Ω crucial assumption MCP: embedding R(Ω) D(Ω) L (Ω) compact Sobolev spaces for rot = curl R(Ω) := {E L (Ω) : rot E L (Ω)} R 0 (Ω) := {E R(Ω) : rot E = 0} R(Ω) := C R(Ω) (Ω) = {E R(Ω) : τe = 0} R 0 (Ω) := R(Ω) R 0 (Ω) ( = H(curl; Ω) ) analogously for div D(Ω), D 0 (Ω), D(Ω), D 0 (Ω) ( = H(div; Ω) ) and Dirichlet resp. Neumann fields H D (Ω) resp. H N (Ω) H D (Ω) := R 0 (Ω) D 0 (Ω) (finite dimensional by compact embedding) = {E L (Ω) : rot E = 0, div E = 0, τe = 0}
Estimates for the Maxwell Constants open problem: estimates for the Maxwell constants c m in 3D? E R(Ω) D(Ω) H D (Ω) E L (Ω) ( cm,t rot E L (Ω) + div ) / E L (Ω) E R(Ω) D(Ω) H N (Ω) E L (Ω) ( cm,n rot E L (Ω) + div ) / E L (Ω) question: in D well known c p c m,t, c m,n c p with Poincaré constants? c m,t, c m,n? even c p < c m,t = c m,n = c p u H (Ω) u L (Ω) c p u L (Ω) ( c p d, if Ω bd in one dir., trivial) u H (Ω) R u L (Ω) cp u L (Ω) (c p diam(ω)/π, if Ω bd & convex, main result Theorem ( 3 DP) Ω R 3 bounded & convex 60 Payne & Weinberger) c p c m,t c m,n = c p diam(ω)/π note always cp = λ < µ = c p
Step : Problem Reduction by Helmholtz Decomposition (as usual) reminder E R(Ω) D(Ω) H D (Ω) E L (Ω) ( cm,t rot E L (Ω) + div ) / E L (Ω) E R(Ω) D(Ω) H N (Ω) E L (Ω) ( cm,n rot E L (Ω) + div ) / E L (Ω) Helmholtz decomposition splits problems into 4 nicer problems E D(Ω) R 0 (Ω) H D (Ω) }{{} E L (Ω) c m,t,div div E L (Ω) = H (Ω) E R(Ω) D 0 (Ω) H D (Ω) }{{} =rot R(Ω) E D(Ω) R 0 (Ω) H N (Ω) }{{} = H (Ω) E R(Ω) D 0 (Ω) H N (Ω) }{{} E L (Ω) cm,t,rot rot E L (Ω) E L (Ω) c m,n,div div E L (Ω) E L (Ω) cm,n,rot rot E L (Ω) =rot R(Ω)
How to do Step (Helmholtz Decomposition) e.g. tangential case Helmholtz decomposition L (Ω) = H (Ω) D 0 (Ω) }{{}}{{} {}}{{}}{ = R 0 (Ω) rot R(Ω) L (Ω) = H (Ω) H D (Ω) rot R(Ω), H D (Ω) = R 0 (Ω) D 0 (Ω) pick some E R(Ω) D(Ω) H D (Ω) E = E E rot ( H (Ω) D(Ω) ) ( rot R(Ω) R(Ω) ) as well as rot E rot = rot E and div E = div E E L (Ω) = E L (Ω) + Erot L (Ω) c m,t,div div E L (Ω) + c m,t,rot rot Erot L (Ω) max{c m,t,div, c m,t,rot} ( div E L (Ω) + rot E L (Ω) )
Step : First Results reminder E R(Ω) D(Ω) H D (Ω) E L (Ω) ( cm,t rot E L (Ω) + div ) / E L (Ω) E R(Ω) D(Ω) H N (Ω) E L (Ω) ( cm,n rot E L (Ω) + div ) / E L (Ω) E D(Ω) H (Ω) E R(Ω) rot R(Ω) E D(Ω) H (Ω) E R(Ω) rot R(Ω) E L (Ω) c m,t,div div E L (Ω) E L (Ω) cm,t,rot rot E L (Ω) E L (Ω) c m,n,div div E L (Ω) E L (Ω) cm,n,rot rot E L (Ω) trivially: c m,t,rot, c m,t,div c m,t and c m,n,rot, c m,n,div c m,n trivially: c m,t max{c m,t,rot, c m,t,div } and c m,n max{c m,n,rot, c m,n,div } (Helmholtz) trivially: c m,t = max{c m,t,rot, c m,t,div } and c m,n = max{c m,n,rot, c m,n,div } Lemma c m,t,div = c p c m,n,div = c p c m,t,rot = c m,n,rot remains to estimate c m,rot := c m,t,rot = c m,n,rot
Step 3: Main Results reminder E R(Ω) D(Ω) H D (Ω) E L (Ω) ( cm,t rot E L (Ω) + div ) / E L (Ω) E R(Ω) D(Ω) H N (Ω) E L (Ω) ( cm,n rot E L (Ω) + div ) / E L (Ω) E D(Ω) H (Ω) E R(Ω) rot R(Ω) E D(Ω) H (Ω) E R(Ω) rot R(Ω) E L (Ω) c p div E L (Ω) E L (Ω) cm,rot rot E L (Ω) E L (Ω) cp div E L (Ω) E L (Ω) cm,rot rot E L (Ω) trivially: c m,t = max{c m,rot, c p} and c m,n = max{c m,rot, c p} remains to estimate c m,rot Theorem ( 3 DP) Let Ω be bounded and convex. Then c m,rot c p. Moreover, c p c m,t c m,n = c p. equivalent formulation for eigenvalues
Proof of First Theorem Proof... by some functional analysis... A : D(A) H H lin., dens. def., closed with adjoint A : D(A ) H H assume D(A) R(A ) H compact! note H = R(A ) N(A) [ ] [ 0 A A define M := and note M A 0 := ] A 0 AA M, M, A A, AA self-adjoint with compact resolvent pure point spectra σ p(m) = ± σ p(a A) = ± σ p(aa ) = ±{κ, κ,...}, 0 κ n looking at first resp. second eigenvalues Lemma (school of Rolf Leis: R. Leis & R. Picard, N. Weck, K.-J. Witsch,... ) Au A v H inf H 0 u D(A) R(A ) u = inf 0 v D(A ) R(A) v H H
Proof of First Theorem... reminder A : D(A) H H lin., dens. def., closed, adjoint A : D(A ) H H D(A) R(A ) H compact (R(A ) = N(A) and R(A), R(A ) closed and u D(A) R(A ) : u H c Au H ) Au A v H inf H 0 u D(A) R(A ) u = inf 0 v D(A ) R(A) v H H especially A := : H (Ω) L (Ω) L (Ω), A := div : D(Ω) L (Ω) L (Ω) R(A ) = N(A) = {0} = L (Ω) H (Ω) L (Ω) = H (Ω) L (Ω) compact by Rellich s selection theorem c p = λ = inf 0 u H (Ω) c m,t,div = c p known! u L (Ω) u L (Ω) = inf 0 E D(Ω) H (Ω) div E L (Ω) E L (Ω) = c m,t,div
Proof of First Theorem... reminder A : D(A) H H lin., dens. def., closed, adjoint A : D(A ) H H D(A) R(A ) H compact (R(A ) = N(A) and R(A), R(A ) closed and u D(A) R(A ) : u H c Au H ) Au A v H inf H 0 u D(A) R(A ) u = inf 0 v D(A ) R(A) v H H especially A := : H (Ω) L (Ω) L (Ω), A := div : D(Ω) L (Ω) L (Ω) R(A ) = N(A) = R H (Ω) R H (Ω) L (Ω) compact by Rellich s selection theorem c p u L = µ = inf div E (Ω) L 0 u H (Ω) R u = inf (Ω) L (Ω) 0 E D(Ω) H E = (Ω) L c (Ω) m,n,div c m,n,div = c p known!
Proof of First Theorem... reminder A : D(A) H H lin., dens. def., closed, adjoint A : D(A ) H H D(A) R(A ) H compact (R(A ) = N(A) and R(A), R(A ) closed and u D(A) R(A ) : u H c Au H ) Au A v H inf H 0 u D(A) R(A ) u = inf 0 v D(A ) R(A) v H H especially A := rot : R(Ω) L (Ω) L (Ω), A := rot : R(Ω) L (Ω) L (Ω) R(A ) = rot R(Ω) R(Ω) rot R(Ω) R(Ω) D(Ω) L (Ω) compact by MCP c m,t,rot = κ = inf 0 E R(Ω) rot R(Ω) rot E L (Ω) E L (Ω) rot E L = inf (Ω) 0 E R(Ω) rot R(Ω) E L (Ω) = c m,n,rot c m,t,rot = c m,n,rot := c m,rot remains to estimate only one constant c m,rot!
Proof of Second (Main) Theorem Proof crucial estimate for convex domains Lemma ( 98 C. Amrouche, C. Bernardi, M. Dauge, V. Girault) Ω R bd and convex. Then E R(Ω) D(Ω), R(Ω) D(Ω) H (Ω) continuous and E L (Ω) ( rot E L (Ω) + div E L (Ω)). related, earlier, partial results by J. Kadlec ( 64), R. Leis ( 68), P. Grisvard ( 7, 85), J. Saranen ( 8), J.-C. Nedelec ( 8), V. Girault & P.-A. Raviart ( 86), M. Costabel ( 9) pick E R(Ω) rot R(Ω) = R(Ω) D 0 (Ω) H N (Ω) = R(Ω) D 0 (Ω) (Ω convex) E, a L (Ω) = rot H, a L (Ω) = 0 for all a R3 since H R(Ω) E H (Ω) (R 3 ) E L (Ω) c m,rot c p very simple!!! Poincaré c p E L (Ω) Lemma c p rot E L (Ω)
Last Slide! Merci / Thank You more results: also inhomogeneous media, i.e., ε id, µ id non-smooth also ND-case, i.e., Ω R N with differential forms, same result also non-convex polygons (not too pointy) or combinations applications: functional a posteriori error estimates for problems with rot preconditioning in numerical algorithms with rot...